If I uninstall Java before installing Sweet Home 3D, will other applications use the Java that comes with Sweet Home 3D?

No, the Java version that comes with Sweet Home 3D isn't registered in the system.

When there is a new version of Java available (usually for security updates), is there anything related to Sweet Home 3D that I need to know?

The Java version available with Sweet Home 3D installer won't be updated.
Note it's not a modified version of Java, so a hacker who knows where Sweet Home 3D is installed and how it's installed, could use the Java Virtual Machine that comes with Sweet Home 3D. But it's very unlikely.